Method of converting a resource into a product

ABSTRACT

The present invention is concerned in particular with the scheduling of maintenance actions such as washing events for a compressor of a gas turbine. An objective function including fuel/power price forecasts is evaluated/optimised in order to determine the advisability of a washing event. The cost function depends on a state vector comprising both Integer/Boolean and continuous state variables which are interconnected via a set of rules or constraints. Mixed Integer Programming (MP) is a used for implementing the inventive procedure.

FIELD OF THE INVENTION

The invention relates to the field of converting a resource into aproduct via and maintenance scheduling for engines suffering acontinuous degradation in performance, and in particular to thescheduling of maintenance actions for a compressor of a gas turbine.

BACKGROUND OF THE INVENTION

The performance of a gas turbine is subjected to deterioration due tocompressor fouling and corrosion, inlet filter clogging, thermalfatigue, and oxidisation of hot gas path components. The performancedeterioration results in loss in power output and/or increase in fuelconsumption and impacts both revenues and equipment life cycle costs.

The performance degradation attributed to compressor fouling is mainlydue to deposits formed on the blades of the first compressor stages byparticles carried in by the air that are not large enough to be blockedby the inlet filter. These particles may comprise sludge or pollen inrural areas, dust, rust and soot particles or hydrocarbon aerosols inindustrials areas, salt in coastal areas or simply water droplets. Thedeposits result in a reduction of compressor mass flow rate, efficiency,and pressure ratio. As about half of the energy contained in the fuelburned by the gas turbine is consumed by the compressor, a noticeableincrease in fuel consumption has to be accepted in order to maintain aconstant power output.

Compressor fouling is a recoverable degradation that can be alleviatedby periodic on-line or off-line compressor washes. In an on-line wash,distilled or at least demineralised water is injected into thecompressor while the gas turbine is running. Complete performancerecovery can only be achieved by an off-line wash (requiring plantshutdown) where distilled water, together with a detergent, is sprayedinto the gas turbine and stays in contact with the compressor blades andvanes. Currently, the washing schedule is made manually by the utilityoperator and the washing is typically scheduled in connection with otherplanned shutdowns. Alternatively, the washing is scheduled in the slackperiods, when the revenues from electricity sales are low.

Inlet filter clogging reduces the gas turbine air flow and compressorinlet pressure and thus adversely affects gas turbine performance.Replacing the old filter with a new or cleaned one can recover the lostperformance. However the performance degradation associated withfrictional wear and/or concerning hot gas path components is referred toas non-recoverable, the only remedy being an engine overhaul.

In the article “Real Time On-Line Performance Diagnostics of Heavy-DutyIndustrial Gas Turbines” by S. C. Gillen et al., Journal of Engineeringfor Gas Turbines and Power, October 2002, Vol. 124, p. 910-921, amaintenance schedule for the compressor washing and inlet filterreplacement balances the maintenance costs against lost revenue andextra fuel costs. The optimal future time to do the washing is foundwhen the integrated cost due to compressor fouling (extra fuel burnedand power lost) equals the costs for the maintenance process. However,neither the evolution of the fuel price in the near future, nor logicalconstraints such as planned outages and part load, are taken intoaccount.

Compressor maps are graphical representations of functions or functionalrelationships relating e.g. the mass flow of a working fluid through acompressor or turbine and/or the efficiency of the compressor or turbineprocess to measured or estimated process states such as temperatures,pressures or speeds. Generally, the manufacturers of the compressors orturbines have sufficient experimental data to estimate e.g. thenon-recoverable efficiency degradation, also known as “guarantee curve”.On the other hand, the turbine operator himself can approximate orestimate the non-recoverable efficiency η_(E), e.g. by means of aninterpolation of a particular process state recorded at the restartfollowing a limited number of off-line washing events.

In the European Patent Application 02405844.8 a method based on StateAugmented Extended Kalman Filtering techniques is disclosed which allowsto obtain and continuously update estimates of the above compressor mapsor functions on-line, i.e. during process operation. In the KalmanFilter, the computed output is compared with the measured outputs andthe actual (i.e. taking into account recoverable degradation) efficiencyη as an augmented state (parameter estimate) is updated. One importantaspect in this procedure is the correction of all measured data tostandard temperature and pressure conditions for dry air. Despite of thefact that the load level may change several times in-between two washingevents, the foregoing procedures generally assume the turbine to workconstantly on full or base load and do not take into account part load.In the absence of any maintenance action, both the actual efficiency ηand the estimated efficiency η_(E) follow an exponential law trend inoutput degradation. An eventual saturation or levelling off is assumedto be due to the stabilization of the thickness and shape of the bladedeposits.

More generally, any system comprising engines or other pieces ofequipment that suffer from a continuous degradation in efficiency can beat least temporarily relieved by maintenance actions. However, thescheduling of the latter is not a straightforward task if time-dependentconstraints influence on the optimal timing. This is the case in asystem that converts a resource into a product where both the resourceand the product are each attached different time-dependent properties.These properties are normalized and equivalent to an objective quantityper unit of measure. In the case of a gas turbine as outlined above, thenormalized properties are costs or prices per unit of mass or energy(i.e. per kg or per MWh) for the fuel and the electricity generated.

In a different case, the system may comprise a generator and otherequipment for producing electrical energy from renewable energy sourcessuch as the sun, wind or water, which are all intermittent ortime-dependent by nature. The normalized properties in this case are thenatural power, i.e. the energy per unit of time delivered by theresource, and the electrical power produced according to a demand by oneor a plurality of consumers.

Because of the decreasing efficiency and the time-dependent normalizedproperties, the objective quantity introduced above is the subject of acertain balance between the resource end and the product end in theconversion process. There is generally a difference between the amountof the objective quantity entering the system and the amount leaving thesystem. Correspondingly, the objective quantity may accumulate at thesystem, or equally, be diverted and used for other purposes than thebasic conversion.

DESCRIPTION OF THE INVENTION

It is therefore an object of the invention to maximize, over apredetermined time horizon, the accumulation of an objective quantity ata system converting a resource into a product as described above. Theseobjectives are achieved by a method for converting a resource into aproduct according to claim 1, a computer program for scheduling amaintenance action for an engine according to claim 1 as well as amethod for scheduling a compressor washing event for a compressor of agas turbine according to claim 12. Further preferred embodiments areevident from the dependent patent claims.

The performance of an engine or other piece of equipment can be improvedby a single or a succession of maintenance actions, thus increasing theefficiency of a system comprising the engine and converting or refininga primary resource into a product. An expenditure for the maintenanceevent and forecasts for the evolution of a time-dependent property ofboth the resource and the product over a predetermined time horizon aretaken into account and constitute the main ingredients of an objectivefunction representing the change in accumulation of an objectivequantity at the system due to a continued degradation in performance ofthe engine. The objective function further depends on at least one statevariable related to a maintenance event at a first time step, and isminimized or solved with respect to this state variable in order todetermine the advisability and/or type of a maintenance action. Thus, aflexible maintenance scheduling approach is provided, where a futureevolution of the respective properties of the resource and the productinfluences the decision at present, and from which optimised maintenanceschemes or plans covering an arbitrary time span are obtained.

In a first embodiment of the invention, the system converts naturalpower from an intermittent renewable energy source such as the sun, windor water, into electric power to meet a time-dependent demand inelectric power. The objective quantity in this case is energy, and thenormalized property per unit time corresponds to the aforementionednatural or electric power. Likewise, the maintenance expenditure isexpressed in the same physical units as the objective quantity, i.e. asan amount of energy that has to be spent and thus has to be consideredin the overall energy balance. Maximizing the accumulation of theobjective quantity according to the invention in this case is equivalentto minimizing the amount of renewable energy needed for conversion andmaintenance in order to meet the abovementioned demand in electricalpower.

In a second embodiment of the invention, the system converts a more orless permanently available fuel such as gas or oil and represented by atime-dependent disbursement into electricity represented by atime-dependent revenue for the product. The objective quantity in thiscase are costs, and the change in accumulation of the objective quantityat the system are additional system costs.

In a first preferred variant of the second embodiment of the invention,the maintenance planning takes into account a cost-forecast of aquantitative measure for the product, i.e. the planned output based on afuture demand of the product. Hence, fluctuating outputs as well as zerooutputs implied by additional constraints such as a system shut-down ora (non-)availability of a maintenance tool or team can be considered ina straightforward way.

In a second preferred variant of the second embodiment of the invention,the objective function involves a sum over the costs at individualfuture time steps as well as corresponding state variables. Theminimization procedure then covers all these state variables at the sametime. Thus the impact of future maintenance actions at later time stepsis inherently taken into account when evaluating the advisability of amaintenance action at present time.

In addition, maintenance actions can be hard constrained, i.e. thecorresponding state variables are set manually and are not determinedvia the general minimization procedure. Such a predetermined constraintis based on the knowledge of a planned system shut-down or a(non-)availability of a maintenance tool or team at a particular futuretime step (e.g. on the following Sunday), and its impact for earliertime steps preceding the constraint can be considered in astraightforward way.

In a further preferred embodiment of the second embodiment of theinvention, two or more different types of maintenance actions areprovided and represented by two or more corresponding state variables ofpreferably Boolean or Integer type. Different costs and performancebenefits associated with the plurality of maintenance actions greatlyenhance the flexibility of the inventive method and potential savings.

The minimisation procedure of the objective function with respect to anextended state vector comprising both Integer/Boolean state variablesand continuous variables has to respect certain rules or constraintsinterrelating the variables. Mixed Integer Linear Programming (MILP) isthen preferably employed for implementing and carrying out theoptimisation procedure.

The degrading performance of the engine is approximated via a linearmodel for an efficiency measure in combination with corresponding rulesor constraints. Even a intrinsically nonlinear behaviour of adegradation can be captured without introducing too much complexityduring the mathematical/digital implementation of the optimisationprocedure.

The inventive method is suited in particular for scheduling the washingevents of a compressor of a gas turbine.

BRIEF DESCRIPTION OF THE DRAWINGS

The subject matter of the invention will be explained in more detail inthe following text with reference to preferred exemplary embodimentswhich are illustrated in the attached drawings, of which

FIG. 1 shows the degradation of compressor efficiency and turbineoutput,

FIG. 2 is an example of a cost/revenue forecast,

FIG. 3 depicts the evolution of the compressor efficiency according tothe invention, and

FIG. 4 is a calendar with scheduled washing events.

DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS

In the following detailed description, a compressor of a gas turbine isthe preferred engine in need of regular maintenance actions. Asmentioned initially, the inlet filter of this compressor constitutesanother such engine having to be replaced according to a certainschedule. The proposed method may equally well be applied to gaspipelines comprising several compressors along the pipeline. Thesecompressors increase the pressure of the gas, resulting an increasedmass transport and increased revenues. However, in order to maintain themaximum pressure, the compressors need to be overhauled periodically,generating costs and down-time. The optimum maintenance scheme for thesecompressors can be determined according to the same general principlesas outlined below in the case of a compressor of a gas turbine.Accordingly, it is to be understood that the conversion process referredto includes not only a conversion of the resource into a materiallydifferent product, but also a simple transportation of the resource, thedifference in cost and price in this case resulting from geographicalimbalances.

In order to illustrate the underlying problem, FIG. 1 depicts thecompressor efficiency η of a gas turbine as a function of time t andaccording to an exemplary compressor washing schedule including anoff-line washing at t₁ and an on-line washing at t₃. In particular,curve 1 represents the non-recoverable degradation (in the absence ofany blade deposits) and curve 2 represents the recoverable degradation(without any washing at all) of the compressor efficiency. Curve 3depicts the “real” efficiency according to the exemplary washingschedule, comprising three distinct sections separated by the twowashing events. Efficiency curve 3 first follows curve 2, and at the endof the off-line washing at t₂, momentarily climbs to curve 1 beforedegrading again. On the other hand, the gain in efficiency following anon-line washing between t₃ and t4 is less pronounced, i.e. curve 1 isnot attained at time t₄.

At the same time, the power output P at constant fuel rate, i.e.provided the operator does not burn more fuel in order to compensate forany loss in efficiency in order to fulfil his contracts, follows asimilar trend as the efficiency curves. During the off-line washing, nooutput power is produced, i.e. the hatched energy area A2 is “lost” interms of revenues. However, following the first wash, the output poweris increased, i.e. the area A3 is gained. During the on-line washbetween t₃ and t₄, power is still produced at part load (energy areaA4), but the gain (area A5) is less pronounced. Comparing the energyareas allows to crudely approximate the economical impact of the washingevents. However, in reality, the optimisation procedure has to considerother costs associated with the washing, such as the consumption of thewashing chemicals and the usage of material due to the shutdown andstartup operations, e.g. the equivalent operating hours (EOH) associatedtherewith.

The optimisation problem includes the following variables:

-   t_(i)=Time (e.g. hours) at optimisation step i.-   C₁(t_(i))=Predicted cost ($) of chemicals, water and energy as    well-as predicted equipment lifetime costs related to off-line    washing at time t_(i)-   C₂(t_(i))=Predicted cost ($) of chemicals, water and energy as well    as predicted equipment lifetime costs related to on-line washing at    time t_(i)-   δ₁(t_(i))=Boolean variable. Equals 1 if off-line washing made at    time t_(i), 0 otherwise-   δ₂(t_(i))=Boolean variable. Equals 1 if on-line washing made at time    t_(i), 0 otherwise-   P_(PWR)(t_(i))=Planned or forecasted power output (MW) at time t_(i)-   P_(Price)(t_(i))=Predicted sales price ($/MWh) for produced or    acquired power at time t_(i)-   D₁(t_(i))=Predicted duration (hours) of an off-line washing at time    t_(i)-   D₂(t_(i))=Predicted duration (hours) of an on-line washing at time    t_(i)-   C_(fuel)(t_(i))=Predicted fuel costs ($/kg) at time t_(i)-   Δf_(fuel)(t_(i),η(t_(i)),P_(PWR)(t_(i)))=Modelled additional fuel    flow (kg/hour) due to degradation as a function of efficiency η and    power output at time t_(i)-   η_(E)(t_(i))=Predicted expected (isentropic) efficiency at time    t_(i)-   η(t_(i))=Predicted actual (isentropic) efficiency at time t_(i)

The expected efficiency η_(E) is the efficiency that a clean compressorwould yield and which can only be achieved after an off-line washing.The degradation of η_(E) is illustrated by the slowly decreasing curve 1in FIG. 1, and cannot be recovered simply by a washing process. Theactual efficiency η is the expected efficiency η_(E) minus a recoverableperformance degradation since the last washing.

Constraints representing e.g. real physical or logical facts interrelatethe abovementioned variables and/or translate into correspondingboundary conditions or rules. By way of example, if there is a plannedplant shutdown at time ti, the planned power output P_(PWR)(t_(i)) willbe zero. Furthermore, as on-line and off-line washing cannot be made atthe same time step, the following rule R1 will be needed:0>=δ₁(t _(i))+δ₂(t _(i))>=1.   (R1)

An instantaneous cost function at time step ti can now be defined asfollows:C(t _(i))=C ₁(t _(i))δ₁(t _(i))+(P _(PWR)(t _(i))D ₁ P _(Price)(t_(i))δ₁(t _(i))+C ₂(t _(i))δ₂(t _(i))+(ΔP _(PWR)(t _(i))D ₂ P _(Price)(t_(i)))δ₂(t _(i))+C _(fuel)(t _(i))f _(fuel)(t _(i))(t _(i+1) −t_(i))(1−δ₁)

The first and third terms represent the costs in resources of anoff-line or on-line washing event, respectively. The second and forthterms represent the cost due to the “lost” output power, or thedisbursements of the operator for power acquired from a second source.ΔP_(PWR) indicates the possiblity of a reduced, albeit non-zero, poweroutput during on-line washings. The fifth term equals the fuel costs dueto continued operation, based on the power P_(PWR) to be delivered andthe actual efficiency η via the modelled fuel flow f_(Fuel).

The benefits of a compressor washing is the ability to produce thefuture power output with less fuel. A prospective cost function takinginto account the effect of a washing at time t_(i) in future time stepst_(j) is defined as follows.${R\left( t_{i} \right)} = {\sum\limits_{t_{j}}^{N_{j}}{{C_{fuel}\left( t_{j} \right)}\Delta\quad{f_{fuel}\left( {t_{j},{\eta\left( t_{j} \right)},{P_{PWR}\left( t_{j} \right)}} \right)}\left( {t_{j + 1} - t_{j}} \right)}}$

This prospective cost function relies on the extra fuel flow due to thedifference between the actual compressor efficiency η and thenonrecoverable efficiency η_(E). The time steps t_(j) have to becarefully chosen and generally comprise only a limited number of terms.

An objective function F as the sum of the two aforementioned terms isthen subject to an optimisation criterion which aims at minimising theoverall costs over a predefined and limited time horizon N_(i)≧1, i.e.:${F\left( {\delta_{1},\delta_{2}} \right)} = {\sum\limits_{i}^{N_{i}}\left( {{R\left( t_{i} \right)} + {C\left( t_{i} \right)}} \right)}$min [F(δ₁, δ₂)]

Taking into account all relevant rules, constraints or boundaryconditions, the Boolean variables δ₁(t_(i)) and δ₂(t_(i)), i=1 . . .N_(i), are to be solved for and the maintenance schedule set upcorrespondingly. The indices i and j as well as the number of termsN_(i), N_(j) in the above expressions for F and R, respectively, aregenerally different from one another. If the model for fuel flow iscalculated on an hourly basis (j-index in the prospective cost functionR) while the optimisation problem is computed on a daily basis (i-indexin the objective function F), then the sum for R will typically containN_(j)=24 elements.

Preferably, from the optimisation procedure outlined above, only theoptimal solution for the first time step at t₁ (i.e. δ₁(t₁) and δ₂(t₁))is retained and all further solutions at chronologically later timesteps t_(i), i=2 . . . N_(i) are disregarded. Subsequently and/orfollowing the execution of a maintenance action according to thesolution at to, costs and price forecasts are readjusted and degradationvalues/rates are corrected if possible/necessary. The time horizon isshifted one step forward (hence the sum for the objective function Fstill comprises N_(i) terms) and the optimisation problem is repeatedwith the actual efficiency η at t₂ as a starting value. Advantageously,the procedure may be repeated in advance for e.g. every day of a month,and a maintenance schedule for a longer period of time may thus begenerated.

In the following, further details, aspects and/or simplifications of theinventive method will be presented.

In order to estimate the dependence of the fuel flow f_(fuel) on theefficiency and thus to compute the benefits in fuel savings after awashing, the following equations are needed:$f_{fuel} = {\frac{1}{C_{R}}\quad\left( {{Power} + P_{PWR}} \right)}$${Power} = {{\left( {T_{a\quad c} - T_{amb}} \right)f_{a}{C_{a}\left( {T_{a\quad c} - T_{amb}} \right)}} = {\frac{1}{\eta_{is}}\left( {T_{is} - T_{amb}} \right)}}$where Power is the power consumed by the compressor (MW), P_(PWR) is thetotal output power from the plant (MW), C_(g) is the fuel capacity(MWh/kg), C_(a) is the specific heat of the compressor air (MWh/(kgK)),f_(a) is the massflow of the compressor air, T_(ac) and T_(amb) are thetemperature at the compressor outlet and inlet, respectively, and η_(is)is the compressor isentropic efficiency.

If an offline washing is made, the efficiency increases from η to η_(E).As the compressor degradation due to dirt particles on the compressorblades is typically in the order of 1-2%, the fuel benefit model assumesthat η and η_(E) do not differ greatly and that in a firstapproximation, f_(a) and T_(is) are the same for η and η_(E). It isfurther assumed that P_(PWR) is constant and independent of thecompressor efficiency. Hence,${\Delta\quad f_{fuel}} = {\frac{1}{C_{g}}f_{a}{C_{a}\left( {T_{is} - T_{amb}} \right)}{\Delta\left\lbrack \frac{1}{\eta_{is}} \right\rbrack}}$

In a second approximation, a nominal efficiency variable η₀ known frome.g. standard compressor maps and a constant γ(0<γ<1) describing theeffectiveness of the online washing operation (γ=1 for offline washing)are introduced, and the final fuel benefit model is then given by.${\Delta\quad f_{fuel}} \approx {\frac{1}{C}{Power}*\frac{\gamma}{\eta_{0}}\left( {\eta_{E} - \eta} \right)}$

Both the actual efficiency η and the estimated efficiency η_(E) dodecrease in an exponential-like manner before eventually levelling off.Nevertheless a linear model of efficiency degradation as a function oftime and of on/off-line washings, introducing new help variables α_(r),α_(n), is capable of reproducing the nonlinear behaviour of thedegradation. The parameters of this linear degradation model ε_(r),ε_(n), γ, α_(r)(0), α_(n)(0) must be tuned in such a way that themodelled degradation curves for η_(E) and η then match the realdegradation curves of the compressor under consideration and may beupdated frequently.

With the following notation

-   α_(r)(0)=Initial degradation level (recoverable)-   α_(n)(0)=Initial degradation level (non-recoverable)-   α_(r)(t_(i))=recoverable degradation at time i-   α_(n)(t_(i))=non-recoverble degradation at time i-   ε_(r)=degradation rate (recoverable),-   ε_(n)=degradation rate (non-recoverable)-   γ=on-line washing effectiveness (γ<1)

the degradation of the continuous state variables/efficiencies aremodelled by the following rulesη(t _(i+1))=η(t _(i))−α_(r)(t _(i))η_(E)(t _(i+1))=η_(E)(t _(i))−α_(n)(t _(i))   (R2)

with the degradation rates (non-negative by definition)α_(r)(t _(i+1))=α_(r)(t _(i))−ε_(r)α_(n)(t _(i+1))=α_(n)(t _(i))−ε_(n)   (R3)

In the event of a washing process at t_(i), a predicted increase inisentropic efficiency is modelled by replacing the previously calculatedvalues as follows:δ₁(t _(i))=1→η(t _(i))=η_(E)(t _(i)) α_(n)(t _(i))=α(0)δ₂(t _(i))=1→η(t _(i))=η(t _(i))+γ(η_(E)(t _(i))−η(t _(i))) α_(n)(t_(i))=α(0)   (R4)

In a more refined version of the inventive method known as “hybridsystem model”, an extended state vector x including, apart from thewashing states δ₁ and δ₂ as introduced above, further Boolean states δ₃and δ₄ representing a normal working state and an idle staterespectively. In addition, the non-integer or continuous variables η andη₂ for current and recoverable efficiency, and z₁, z₂, z₃ for the costof online washing, offline washing and the cost of extra fuel due todegradation, are introduced. The objective function becomesF(x)=z₁+z₂+z₃, with x=(δ₁, δ₂, δ₃, δ₄, η, η₂, z₁, z₂, z₃) the extendedstate vector.

The above rules R2, R3, R4 for η and η_(E) remain valid, while theoriginal rule R1 is to be replaced by the modified rule R1′ of the formδ₁+δ₂+δ₃+δ₄=1. As above, z₁=δ₁C₁ and z₂=δ₂C₂, whereas the rule for z₃ issignificantly simplified and takes the linear form known from the fuelbenefit model above, i.e. z₃=P₅ (η−η₂), where P₅ is the product of thefuel price, the time step, and the constants in the expression forΔf_(fuel) derived above. In addition, the time steps of the optimisationproblem and the fuel benefit model are preferably assumed to be of equallength and identical to one day.

The decision problem is formulated as a receding horizon optimisation(also known as Model-Predictive Control). The Integer/Boolean andcontinuous variables are comprised in a state vector x, which is thenduplicated for each time step t_(j) within a predetermined finite timehorizon. In addition, the aforementioned rules R1 . . . R4 as well asthe objective function F are duplicated for each time step. In thesubsequent Mixed Integer Programming (MIP) approach, only a limitedamount of N time steps ahead are considered. The optimal solution forthe first time step at t_(i) is retained and executed by the plantoperator. After the execution has completed, the plant operatorreadjusts costs and price forecasts and corrects degradationvalues/rates if possible/necessary, shifts the time horizon one stepforward and repeats the optimisation problem with the actual efficiencyη at t_(i) as a starting value. If a maintenance schedule for a longerperiod of time is required, the procedure/operator will iterate throughthe calendar day by day, shifting the time horizon one step forward andretaining only the optimal solution for the first time step in eachiteration. Again, the latter is recycled as the initial value for thenext iteration step.

The procedure outlined in the foregoing involves both integer (such asδ) and continuous (such as η or the cost variables z) state variables,and is therefore based on a hybrid or mixed logical dynamical systemmodel. In addition to the optimisation function, the abovementionedrules, constraints or boundary conditions have to be observed. As longas both of them are linear, as e.g. the fuel benefit model and theapproximated efficiency degradation disclosed above, the procedure iscalled a Mixed Integer Linear Programming (MILP) approach oroptimisation framework.

Given a linear dependence of the objective function F on a state vectorx comprising both integer and continuous variables, a cost vector g canbe defined and the cost function F is rewritten as F(x)=g^(T)x. With theinequalities building up a constraint matrix A, the MILP formulation forthe state vector x is as follows:Min F(x), subject to Ax<b

Alternatively, if the cost function F involves quadratic terms in thestate vector x as represented by a cost matrix Q, Mixed IntegerQuadratic Programming (MIQP) might be applied. The objective functiontakes the form F(x)=x^(T)Qx+g^(T)x.

The cost matrix Q, cost vector g and the constraint matrix A are fed toa robust and reliable optimisation problem solver, such as e.g. thecommercially available optimiser for solving linear, mixed-integer andquadratic programming problems called CPLEX(http://www.ilo,.com/products/cplex/). Alternatively, if theoptimisation function F and/or the constraints were allowed to involvegeneral nonlinear terms in the state vector x, mixed integer nonlinearprogramming (MINLP) might be applied.

The availability of reliable forecasts for the prices of power (to besold) and fuel (to be purchased) is essential for the successfulimplementation of the method. Such price forecasts can be boughtcommercially from various suppliers (http://www.bmreports.com), andexample being depicted in FIG. 2. On the top graph, a fuel priceforecast C_(fuel) is shown, and on the bottom graph, a power priceforecast P_(Price) ($/MWh) for one year.

The benefit of the inventive method is visible from FIG. 3. The topcurve 1 shows the non-recoverable efficiency η_(E), while the bottomcurve 2 shows the efficiency degradation when every day is hardconstrained to normal operation and no washing at all is foreseen. Themodel converges to an efficiency difference of about 3% between the twosteady-state efficiency levels. The objective function with everydaywashing amounts to a certain amount of money spent mainly on chemicals,whereas without any washing these costs roughly double. This amountcorresponds to the extra money spent on fuel that the plant owner mustpay due to the difference of compressor efficiency compared to thenonrecoverable efficiency level. The curve 3 in the middle shows thesame resulting efficiency when all the hard constraints are removed andthe system is determining the frequency and type of washing events,resulting in an increased online washing frequency when the fuel pricesare high (around day 200). The minimized objective function in this casecomprises a combination of outlays on washing detergents and on extramoney spent on fuel due to degradation. Because the cost of an offlinewashing is very high due to the loss of electricity sales, no offlinewashings is made in this example.

FIG. 4 finally shows an optimised washing schedule for the month ofApril 2003. The days marked with a tick (√) stand for normal operationof the system, whereas the symbol Ø signifies online washing. Theschedule shows that the optimal washing frequency is every 2nd or 3rdday, depending on the fuel prices. Again, no offline washing isenvisaged, as the costs associated with offline washing are so high (dueto lost power sales) that it is never economical to shut down the plantonly for the washing. However, the utility owner can hard-constrain theoptimiser to do offline washings on specific calendar days when theplant is shut down for other reasons, for example maintenance of thegenerator.

To summarize, in the inventive method of scheduling a maintenance actionfor an engine suffering from a degradation in performance, wherein theengine is part of a system converting a resource into a product, andwherein the maintenance action increases the performance of the engine,the following steps are perfomed:

-   a) introducing a state variable (δ₁) representing a maintenance    action at a first time step (t₁) and an efficiency measure (η)    representing the degrading performance of the engine,-   b) providing a cost-estimate of a price for the maintenance action    (C₁) and an initial value for the efficiency measure (η(t_(i))), and    providing for N_(j)>1 discrete time steps (t_(j)) into the future, a    cost-forecast of a disbursement (C_(fuel)(t_(j))) for the resource    and of a revenue (P_(Price)(t_(j))) for the product,-   c) setting up an objective function F including the additional    system costs related to the degradation in performance, based on the    cost-estimate (C₁), the efficiency measure (η), the cost-forecasts    (C_(fuel)(t_(j)), P_(Price)(t_(j))), and the state variable (δ₁),-   d) minimising the objective function F with respect to the state    variable (δ₁), and scheduling a maintenance action at the first time    step (t₁) accordingly.

In this way, a flexible and economically optimised scheduling approachis provided, where a future evolution of the market prices influencesthe decision at present, and from which optimised maintenance schemes orplans covering an arbitrary time span are obtained.

1. A method of converting a resource into a product, wherein theresource and the product are each represented by a time-dependentnormalized property (P_(IN), C_(fuel); P_(OUT), P_(Price)) equivalent toan objective quantity per unit of measure, wherein a performance of asystem converting the resource into the product is represented by anefficiency measure (η), and wherein the performance is degradingcontinuously and can be increased by a maintenance action applied to thesystem, wherein the method comprises a) introducing a state variable(δ₁) representing a maintenance action at a first time step (t₁) andestimating a maintenance expenditure (C₁) for said maintenance action,b) providing an initial value for the efficiency measure (η(t_(i))), c)providing, for N_(j)>1 discrete time steps (t_(j)) into the future, aforecast of the normalized property (P_(IN)(t_(j)), C_(fuel)(t_(j)))representing the resource and a forecast of the normalized property(P_(OUT)(t_(j)), P_(Price)(t_(j))) representing the product, d) settingup an objective function F including a change in accumulation of theobjective quantity at the system related to the degradation inperformance, and based on the maintenance expenditure (C₁), theefficiency measure (η), the normalized property forecasts(P_(IN)(t_(j)), C_(fuel)(t_(j)); P_(OUT)(t_(j)), P_(Price)(t_(j))), andthe state variable (δ₁), e) minimising the objective function F withrespect to the state variable (δ₁), and scheduling a maintenance actionat the first time step (t₁) accordingly.
 2. The method according toclaim 1, wherein the resource is an intermittent renewable energy sourcerepresented by a time-dependent supply power (P_(IN)) and in thatwherein the product is electricity represented by a time-dependentdemand power (P_(OUT)).
 3. The method according to claim 1, wherein stepa) comprises providing a cost-estimate of a price for the maintenanceaction (C₁), step c) comprises providing a cost-forecast of adisbursement (C_(fuel)(t_(j))) representing the resource and acost-forecast of a revenue (P_(Price)(t_(j))) for the product, and stepd) comprises setting up an objective function F for the additionalsystem cost related to the degradation in performance as the change inaccumulation of the objective quantity at the system.
 4. The methodaccording to claim 3, wherein for the N_(j)>1 discrete time steps(t_(j)) into the future, a forecast of a quantitative measure of theproduct (P_(PWR)(t_(j))) is provided and in that wherein the objectivefunction F is also based thereupon.
 5. The method according to claim 3,the objective function F comprises the costs (C(t_(i))+R(t_(i)); z₄) atN_(i)>1 future time steps (t_(i)), as well as state variables(δ₁(t_(i))) representing each a maintenance action at a correspondingfuture time step (t_(i)), and in that wherein the objective function Fis minimized with respect to all these state variables (δ₁(t_(i))). 6.The method according to claim 5, wherein the minimization procedure isconstrained in the case of a predetermined maintenance action at afuture time step (t_(m)) by setting the corresponding state variable(δ₁(t_(m))) to a fixed value.
 7. The method according to claim 5,wherein at least two different maintenance actions with different impacton the performance of the engine are possible and wherein their presenceor absence is represented by at least two different Boolean statevariables (δ₁, δ₂).
 8. The method according to claim 7, wherein anextended state vector (x) comprises the Boolean state variables (δ₁, δ₂)as well as continuous state variables (η, η₂, z₁, z₂, z₃), and in thatwherein rules (R1, R1′, R2-R4) are interrelating these state variables.9. The method according to claim 7, wherein the degradation inperformance is modelled via a linear model for the efficiency measure(η) resulting in a set of corresponding rules (R2-R4) interrelating theefficiency measure (η) with the Boolean state variables (δ₁, δ₂). 10.The method according to claim 3, wherein the system is a gas turbineconverting gaseous fuels as a resource into mechanical or electricalenergy as a product, and wherein the engine is a compressor suffering adegradation in efficiency (η) and the maintenance action is a compressorwashing event.
 11. A computer program for scheduling a maintenanceaction for an engine suffering from a degradation in performance whichis loadable and executable on a data processing unit and which computerprogram, when being executed, performs the steps according to the methodof converting a resource into a product as claimed in claim
 1. 12. Amethod of scheduling a compressor washing event for a compressorsuffering from a degradation in efficiency (η), wherein the compressoris part of a gas turbine converting gaseous fuels into mechanical orelectrical energy, and wherein the compressor washing event increasesthe efficiency of the compressor, comprising the steps of a) introducinga state variable (δ₁) representing a compressor washing event at a firsttime step (t₁), b) providing a cost-estimate of a price for thecompressor washing event (C₁) and an initial value for the efficiency(η(t₁)), and providing for N_(j)>1 discrete time steps (t_(j)) into thefuture, a cost-forecast of a disbursement (C_(fuel)(t_(j))) for thegaseous fuels and of a revenue (P_(Price)(t_(j))) for the mechanical orelectrical energy, c) setting up an objective function F for theadditional system costs related to the degradation in efficiency, basedon the cost-estimate (C₁), the efficiency (η), the cost-forecasts(C_(fuel)(t_(j)), P_(Price)(t_(j))), and the state variable (δ₁), d)minimising the objective function F with respect to the state variable(δ₁), and scheduling a compressor washing event at the first time step(t₁) accordingly.
 13. The method according to claim 4, wherein thesystem is a gas turbine converting gaseous fuels as a resource intomechanical or electrical energy as a product, and wherein the engine isa compressor suffering a degradation in efficiency (η) and themaintenance action is a compressor washing event.
 14. The methodaccording to claim 5, wherein the system is a gas turbine convertinggaseous fuels as a resource into mechanical or electrical energy as aproduct, and wherein the engine is a compressor suffering a degradationin efficiency (η) and the maintenance action is a compressor washingevent.
 15. The method according to claim 6, wherein the system is a gasturbine converting gaseous fuels as a resource into mechanical orelectrical energy as a product, and wherein the engine is a compressorsuffering a degradation in efficiency (η) and the maintenance action isa compressor washing event.
 16. The method according to claim 7, whereinthe system is a gas turbine converting gaseous fuels as a resource intomechanical or electrical energy as a product, and wherein the engine isa compressor suffering a degradation in efficiency (η) and themaintenance action is a compressor washing event.
 17. The methodaccording to claim 8, wherein the system is a gas turbine convertinggaseous fuels as a resource into mechanical or electrical energy as aproduct, and wherein the engine is a compressor suffering a degradationin efficiency (η) and the maintenance action is a compressor washingevent.
 18. The method according to claim 9, wherein the system is a gasturbine converting gaseous fuels as a resource into mechanical orelectrical energy as a product, and wherein the engine is a compressorsuffering a degradation in efficiency (η) and the maintenance action isa compressor washing event.